Image Processing

where D′ indicates the break point for the piecewise linear function. A radial cross section of the trapezoid low-pass filter of order 1 is shown in Figure 3.21. Compared to the transfer function of the ideal low-pass filter, it has a transition near the cutoff frequency and thus reduces the ring effect. On the other hand, compared to the transfer function of the Butterworth low-pass filter, its transition is not smooth enough, so its ring effect is stronger than that of the Butterworth low-pass filter.

A 2-D exponential low-pass filter of order n has the transfer function (the cutoff frequency is D0)

$H\left(u,v\right)=\exp \left\{-{\left[D\left(u,v\right)/D_0\right]}^n\right\}\left(3.19\right)$

A radial cross section of the exponential low-pass filter ...